Matrices And Economics

There are many uses for matrices in everyday life. One example of how matrices can be used is with economics.Here is an economic problem that can be solved using matrices:
Consider an open economy with three industries: coal-mining operation,electricity-generating plant and auto-manufacturing plant.To produce Re 1 of coal,the mining peration must purchase Re 0.1 of its town production,Rs 0.30 of electricity and Re 0.1 worth of automobile for its transportation.To produce Re 1 of electricity, it takes Rs 0.25 of coal,Rs 0.4 of electricity and Rs 0.15 of automobile.Finally,to produce Re 1 worth of automobile,the auto-manufacturing plant must purchase Rs 0.2 of coal, Rs 0.5 of electricity and consume Rs 0.1 of automobile. Assume also that during a period of one week,the economy has an exterior demand of Rs 50,000 worth of coal,Rs 75,000 worth of electricity,and Rs 1,25,000 worth of autos. Find the production level of each of the three industries in that period of one week in order to exactly satisfy both the internal and the external demands.

Solution:
The input-output matrix of this economy is
A =
0.1 0.25 0.2
0.3 0.4 0.5
0.1 0.15 0.1

and the demand vector is
d =
50,000
75,000
125,000

Now using the equation
P = (I-A)^-1 . d

where
I - A =
0.9 -0.25 -0.2
-0.3 0.6 -0.5
-0.1 -0.15 0.9

Using the Gauss-Jordan elimination technique, we find that
( I - A )^-1 =
1.464 0.803 0.771
1.007 2.488 1.606
0.330 0.503 1.464

Which gives
P =
1.464 0.803 0.771
1.007 2.488 1.606
0.330 0.503 1.464
x
50,000
75,000
125,000
=
229921.59
437795.27
237401.57

So the total output of the coal-mining operation must be Rs 229921.59, the total output for the electricity-generating plant is Rs 437795.27 and the total output for the auto-manufacturing plant is Rs 237401.57.

In conclusion, linear algebra is not only a course full of theorical studies but it is used to solve many problems of our daily life.


http://www.scribd.com/doc/19613606/Applications-of-matrices-to-Business-and-Economics